list manipulation - Find all disjoint subsets of equal length

Suppose that $A$ is a list with $n$ elements where $n$ is even. I want to write a function that returns all pairs $(A_1,A_2)$ where the sets $A_1$ and $A_2$ each have length $\frac{n}{2}$, $A_1 \cap A_2=\emptyset$, and $A_1,A_2 \subset A$.

Answer

I'm assuming in A_1 A_2 pair, order matters:

set = Range[6];

Transpose[{#, Reverse@#}] & @ Subsets[#, {Length[#]/2}] & @ set

{{{1, 2, 3}, {4, 5, 6}}, {{1, 2, 4}, {3, 5, 6}}, {{1, 2, 5}, {3, 4, 
 6}}, {{1, 2, 6}, {3, 4, 5}}, {{1, 3, 4}, {2, 5, 6}}, {{1, 3, 
5}, {2, 4, 6}}, {{1, 3, 6}, {2, 4, 5}}, {{1, 4, 5}, {2, 3, 
6}}, {{1, 4, 6}, {2, 3, 5}}, {{1, 5, 6}, {2, 3, 4}}, {{2, 3, 
4}, {1, 5, 6}}, {{2, 3, 5}, {1, 4, 6}}, {{2, 3, 6}, {1, 4, 
5}}, {{2, 4, 5}, {1, 3, 6}}, {{2, 4, 6}, {1, 3, 5}}, {{2, 5, 
6}, {1, 3, 4}}, {{3, 4, 5}, {1, 2, 6}}, {{3, 4, 6}, {1, 2, 
5}}, {{3, 5, 6}, {1, 2, 4}}, {{4, 5, 6}, {1, 2, 3}}}

If pair isn't ordered you can take halof of above or spare some memory doing:

With[{l = Length[#]}, 
   Transpose[{#, Reverse@#2}] & @@ 
    Partition[Subsets[#, {l/2}], Binomial[l, l/2]/2]] &@set